Math. Struct. in Comp. Science: page 1 of 36. c Cambridge University Press 2016 doi:10.1017/S0960129516000281 Mackey-complete spaces and power series – a topological model of differential linear logic MARIE KERJEAN and CHRISTINE TASSON Laboratory PPS, Universit´e Paris Diderot, Paris, France Emails:
[email protected],
[email protected] Received 10 July 2015; revised 29 June 2016 In this paper, we describe a denotational model of Intuitionist Linear Logic which is also a differential category. Formulas are interpreted as Mackey-complete topological vector space and linear proofs are interpreted as bounded linear functions. So as to interpret non-linear proofs of Linear Logic, we use a notion of power series between Mackey-complete spaces, generalizing entire functions in C. Finally, we get a quantitative model of Intuitionist Differential Linear Logic, with usual syntactic differentiation and where interpretations of proofs decompose as a Taylor expansion. 1. Introduction Many denotational models of linear logic are discrete, based for example on graphs such as coherent spaces (Girard 1986), on games (Abramsky et al. 2000; Hyland and Ong 2000), on sets and relations, or on vector spaces with bases (Ehrhard 2002, 2005). This follows the intrinsic discrete nature of proof theory, and of linear logic. The computational interpretation of linearity in terms of resource consumption is still a discrete notion, proofs being seen as operators on multisets of formulas. Besides, Ehrhard and Regnier show in Ehrhard and Regnier (2003b) and Ehrhard and Regnier (2003a) how it is possible to add a differentiation rule to Linear Logic, in this way constructing Differential Linear Logic (DiLL).